Why does the world conform to logic?

[Asparagus]

By "does not form a totality," was he saying that R doesn't exist? We're talking about it. We can ask questions about it. We can be either wrong or right in our assessment of it. It has all the hallmarks of an abstract object.

That's what I take him to mean - if "totality" doesn't mean "set" then I cannot parse the meaning of his statement. We can ask questions about "square circles", or about "nonexistent unicorns", but it doesn't mean they exist.

The issue is that set theory leads to a contradiction. R is one example of it. I think Russell was saying that the contradiction is the result of the mistake of treating a collection of classes as one class. But isn't that pretty close to what set theory is doing with natural numbers? I think Russell was questioning the basic assumptions of set theory, not saying that set theory is fine except R can't exist. But I'm totally open to being corrected about that.

But consider the attitude that R can't exist because it's contradictory. Would it follow from that that logic reliably guides us in understanding the world?

[Lomax]

By "does not form a totality," was he saying that R doesn't exist? We're talking about it. We can ask questions about it. We can be either wrong or right in our assessment of it. It has all the hallmarks of an abstract object.

That's what I take him to mean - if "totality" doesn't mean "set" then I cannot parse the meaning of his statement. We can ask questions about "square circles", or about "nonexistent unicorns", but it doesn't mean they exist.

The issue is that set theory leads to a contradiction. R is one example of it. I think Russell was saying that the contradiction is the result of the mistake of treating a collection of classes as one class. But isn't that pretty close to what set theory is doing with natural numbers? I think Russell was questioning the basic assumptions of set theory, not saying that set theory is fine except R can't exist. But I'm totally open to being corrected about that.

I think that's a fair assessment of Russell's approach, and it seems to contain a concession that Russell felt the set could not exist. These days the approach of Principia Mathematica is rarely used, and the mathematical community seems to consider ZFC comparably powerful and more wieldy.

But consider the attitude that R can't exist because it's contradictory. Would it follow from that that logic reliably guides us in understanding the world?

The grammar of your question is odd - am I to assess what follows the fact someone has a particular attitude? We might just as easily take the attitude that R is contradictory because it can't exist. Yes, I think logic generally sharpens our thinking, which doesn't necessary mean that the world conforms to it, rather than the other way around.

[Lomax]

RJG - there have been multiple projects to found mathematics without set theory and it usually leads to a severely crippled system. Your proposed solution (that there can be no such thing as a subset or a superset) is not vastly different from Russell's own Theory of Types, although it goes further.

[Asparagus]

I think that's a fair assessment of Russell's approach, and it seems to contain a concession that Russell felt the set could not exist. These days the approach of Principia Mathematica is rarely used, and the mathematical community seems to consider ZFC comparably powerful and more wieldy.

If Russell says that set theory leads to a contradiction, then it follows that he believed both set theory and the contradiction exist. R is the contradiction, so he believed that R exists. I don't think ZFC is more powerful. It's just contradiction free (as I mentioned before, artificially.)

The grammar of your question is odd - am I to assess what follows the fact someone has a particular attitude? We might just as easily take the attitude that R is contradictory because it can't exist. Yes, I think logic generally sharpens our thinking, which doesn't necessary mean that the world conforms to it, rather than the other way around.

So you believe logic is based on observations?

[Lomax]

I think different logics are based on different things. In general, it is a way of formalising our thinking, for the purposes of scrutiny and rigour. I didn't say ZFC was more powerful. And what you say about Russell and set theory makes no sense. If I believe square circles lead to a contradiction, I do not therefore believe that they must exist and the contradiction is true. I simply throw out the premise.

[Asparagus]

I think different logics are based on different things.

So lets consider the type of logic that appears to come as standard equipment in humans. What would you say it's based upon? If it's observation of the world, that would be in keeping with the view of ancient naturalists who saw logic as immanent in the world. Would you agree or disagree with that?

And what you say about Russell and set theory makes no sense. If I believe square circles lead to a contradiction, I do not therefore believe that they must exist and the contradiction is true. I simply throw out the premise.

"Antinomy" would be the word we'd use to describe set theory (without axioms to exclude contradiction). Square circles don't lead to a contradiction, they are contradictory. There is no premise to throw out. But you're close to the early thinking about set theory: that it would have to be scrapped entirely due to contradiction.

Think of a set of chess rules that lead to a contradiction. Both the rules and the contradiction exist as objects of thought. Once the rules are circulated, they're abstract objects. The fact that contradiction exists as an outcome of execution of the rules doesn't make the rules or the contradiction disappear from existance.

[Positor]

Liar's Paradox
1. Statement T: "This statement is false".
2. Statement S: "This statement, "This statement is false", is false."
3. → Statement S: "This statement, T, is false."
4. → No paradox. No contradiction.

T is not S. Falsely equivocating them as the same, is the logical error.

If T is not S, what exactly do the words "This statement" in T refer to? (On the understanding that T refers only to itself.)

Are you treating all of the following as equivalent, or not?

(a) "This statement is false" is false.
(b) The statement "This statement is false" is false.
(c) This statement, "This statement is false", is false.

[RJG]

If T is not S, what exactly do the words "This statement" in T refer to? (On the understanding that T refers only to itself.)

It refers to the 'entire' statement itself. Therefore, it is this 'entire' statement "T" (and not anything else!) that is false.

T = "This statement is false".
S = "This statement "This statement is false" is false".
→S = The statement T is false.

Note: Statement T and Statement S are two different statements. It is only T (not S) that is false.

Are you treating all of the following as equivalent, or not?

(a) "This statement is false" is false.
(b) The statement "This statement is false" is false.
(c) This statement, "This statement is false", is false.

Yes, these are all equivalent. These are all saying "statement T is false".

[Positor]

T = "This statement is false".
S = "This statement "This statement is false" is false".
→S = The statement T is false.

Note: Statement T and Statement S are two different statements. It is only T (not S) that is false.

1. T is "This statement is false".
2. "This statement" means T.
3. Therefore (from 1 and 2), T asserts that T is false.
4. Therefore (from 1 and 3), T asserts that "This statement is false" (i.e. T) is false.

5. S asserts that "This statement is false" (i.e. T) is false.
6. Therefore (from 4 and 5), T and S assert the same thing.
7. Therefore (from 6), T and S must have the same truth-value(s).

Note also the following:
(a) Both T and S refer to the truth-value of T.
(b) S does not refer to the truth-value of S.
(c) T refers to itself, but S does not.

[RJG]

T = "This statement is false".
S = "This statement "This statement is false" is false".
→S = The statement T is false.

Note: Statement T and Statement S are two different statements. It is only T (not S) that is false.

1. T is "This statement is false".

Yes.

2. "This statement" means T.

No. "This statement" refers to T. "This statement" points to and identifies 'which' statement "is false".

Note: T cannot mean TWO different things. It cannot mean both a) "This statement" AND b) "This statement is false". It is one or the other, but not both.

3. Therefore (from 1 and 2), T asserts that T is false.

No. T is still just T. T is just the statement "This statement is false". It doesn't assert itself is false.

4. Therefore (from 1 and 3), T asserts that "This statement is false" (i.e. T) is false.

No. Again, T is just the statement "This statement is false". Nothing more. It does not assert itself is false.

5. S asserts that "This statement is false" (i.e. T) is false.

No. S asserts "This statement, "this statement is false", is false." Or when reduced/simplified asserts "The statement T is false".

[RJG]

Real paradoxes don't exist any more than 'real magic' does. Those that argue that paradoxes are real, seemingly would likely argue that magic is real. To do so is arguing that the impossible is somehow possible. But, we can't defy logic. Paradoxes and Magic don't exist in reality. There is always a 'trick'!

Liar's Paradox -- "This statement is false" - The 'trick' to solving this paradox, is to first identify what "This" is actually referring to. The "paradox" only exists if one falsely equivocates two different meanings (references) of the word "this".

If "This" refers to the "statement" itself, then
1. Statement T: "statement".
2. Statement S: "This "statement" is false".
3. → Statement S: T is false
4. → No contradiction → No Paradox

If "This" refers to the 'entire' statement "This statement is false", then
1. Statement T: "This statement is false".
2. Statement S: "This statement, "This statement is false", is false".
3. → Statement S: The statement T is false
4. → No contradiction → No Paradox

Russell's Paradox -- "The set of all sets that don't contain themselves" - The 'trick' to Russell's paradox is contained within the first set/statement. It is here where we fall hook, line, and sinker into accepting (believing in) a 'logical impossibility'.

1. Set T: Sets that don't contain themselves.

This is a nonsense statement. This is more nonsensical than saying - the set of "pigs that don't fly". It implies that - sets can somehow contain themselves (...and that pigs can somehow fly). But 'sets' cannot contain themselves. If T is T, then T cannot contain T. Sets can only contain Members (T can only contain ~T's !!).

It is impossible to be both "inside" a box (as a 'member') while "outside" the box (as a 'set'). In other words, it is not logically possible to be in two places at one time. It is one or the other, but never both! Russell's Paradox is just the creation of nonsense, out of nonsense.

[Positor]

2. "This statement" means T.

No. "This statement" refers to T. "This statement" points to and identifies 'which' statement "is false".

Agreed. When I said that "this statement" means T, I did not mean that the words "this statement" constitute T. Obviously they only constitute part of T. But they refer to T, as you say.

3. Therefore (from 1 and 2), T asserts that T is false.

No. T is still just T. T is just the statement "This statement is false". It doesn't assert itself is false.

So, what does it assert as false? "This statement" - but what statement is that, if not T itself?

5. S asserts that "This statement is false" (i.e. T) is false.

No. S asserts "This statement, "this statement is false", is false." Or when reduced/simplified asserts "The statement T is false".

How does that differ from what I say in (5)?

[RJG]

No. T is still just T. T is just the statement "This statement is false". It doesn't assert itself is false.

So, what does it assert as false? "This statement" - but what statement is that, if not T itself?

If I have two apples; one red, one green, sitting on my desk. And I said "This apple is bad". What does "This apple" assert?

1. It 'identifies' (asserts the 'identification' of) the apple/statement (the T statement) that I am talking about (referring to; pointing at).

T = 'green' apple
T = "This statement is false"

2. And it completes the S statement --

S = The apple, T, is bad.
S = The statement, T, is false

5. S asserts that "This statement is false" (i.e. T) is false.

No. S asserts "This statement, "this statement is false", is false." Or when reduced/simplified asserts "The statement T is false".

How does that differ from what I say in (5)?

It doesn't differ. My "No." should be "Yes." We are saying the 'same' thing, but in different words.

[Asparagus]

But 'sets' cannot contain themselves. If T is T, then T cannot contain T.

The set of all non-elephants is not an elephant. There is no intuitive reason to say it can't contain itself. A set is just criteria. It's an abstract object. My guess as to why set theory leads to contradictions is that it has a contradiction in its foundation. Contradiction in the roots, contradiction in the leaves. Something like that.

[RJG]

But 'sets' cannot contain themselves. If T is T, then T cannot contain T.

The set of all non-elephants is not an elephant. There is no intuitive reason to say it can't contain itself.

? "intuitive" to some, ...maybe? - But "logically", it is impossible! Sets cannot contain themselves.

1. 'Sets' can only contain 'Members'.

2. Things can't be on top of themselves, nor inside themselves, nor relative to themselves, nor 'contain' themselves. X is just X. Set X is just Set X

The ONLY possible relationship of X to 'itself', is X=X.

My guess as to why set theory leads to contradictions is that it has a contradiction in its foundation. Contradiction in the roots, contradiction in the leaves. Something like that.

...I don't disagree.

[Asparagus]

? "intuitive" to some, ...maybe? - But "logically", it is impossible! Sets cannot contain themselves.

1. 'Sets' can only contain 'Members'.

2. Things can't be on top of themselves, nor inside themselves, nor relative to themselves, nor 'contain' themselves. X is just X. Set X is just Set X

I had similar confusion when I first started with it. Happy Trails. :)

[Positor]

RJG,

I think we agree about S now, but not about T.

T = 'green' apple
T = "This statement is false"

These are not equivalent in form. "Green apple" is just a subject; it is not a sentence or statement. But "This statement is false" is a sentence and a statement, containing the predicate "is false". It is asserting that something is false.

[RJG]

T = 'green' apple
T = "This statement is false"

These are not equivalent in form. "Green apple" is just a subject; it is not a sentence or statement. But "This statement is false" is a sentence and a statement, containing the predicate "is false".

You are missing the point. This is not about sentences or statements. It is about identifying what "this" is referring to. Whatever "this" refers to, we will call it T, and say it "is false".

It is asserting that something is false.

Yes! So tell me, what 'is' this something that is false??? Whatever this something is, we will call it T, and say it "is false".

1. When someone says -- "This statement is false" -- what does "this" refer to? ...WHICH statement is false? Answer: "This" refers to the (entire) statement "This statement is false", which we will call T. T = "This statement is false". Therefore, "This statement is false", is false! → T is false!

2. When someone says -- "This apple is bad" -- what does "this" refer to? ...WHICH apple is bad? Answer: "This" refers to the apple that is 'green', which we will call T. T = the 'green' apple. Therefore, the 'green' apple, is bad! → T is bad!

[Positor]

1. When someone says -- "This statement is false" -- what does "this" refer to? ...WHICH statement is false? Answer: "This" refers to the (entire) statement "This statement is false", which we will call T. T = "This statement is false". Therefore, "This statement is false", is false! → T is false!

Agreed. So:

1. T states: "This statement is false".
2. "This statement" refers to the entire statement T.
3. Therefore, T asserts that (the entire statement) T is false.

[RJG]

1. T states: "This statement is false".

To be more precise, T = "This statement is false". T does NOT state (or assert) 'itself'!

2. "This statement" refers to the entire statement T.

Yes. (..but to be more precise, "This" refers to 'statement T', not to the "statement" within statement T.)

3. Therefore, T asserts that (the entire statement) T is false.

No. It is "S" that asserts that statement T is false. T cannot assert itself.

The statements S and T (and the "statement" within statement T) are all different statements.

[Positor]

No. It is "S" that asserts that statement T is false. T cannot assert itself.

But I want to know what T (not S) is asserting.

T is a complete statement. It is stating/asserting that something is false (you agreed with this in a recent post). But you say that T cannot assert (anything about) itself; it cannot assert that T itself is false. I cannot reconcile these two positions.

What is T describing as false?

[RJG]

Statement T asserts that the "statement" itself is false. --> U is false.

U: "statement"
T: "This "statement" is false" --> U is false
S: ""This "statement" is false" is false" --> T is false

Example:
U: "1+1=2"
T: ""1+1=2" is false" --> U is false
S: """1+1=2" is false", is false" --> T is false

[RJG]

Bottom-line --

When someone says "This statement is false", what is it that they mean? ...Do they mean:

(a) The "statement" itself is false? - or
(b) The entire statement "This statement is false" is false?

It can ONLY be one OR the other; only (a) OR (b)! ...so WHICH one is it?

Using the variables S and T --

If (a) - If this "statement" (itself) is false, then


1. Statement T: "statement".
2. Statement S: "This "statement" is false".
3. → Statement S: T is false
4. → No contradiction → No Paradox

If (b) - If this (entire) statement "This statement is false" is false, then


1. Statement T: "This statement is false".
2. Statement S: "This statement, "This statement is false", is false".
3. → Statement S: T is false
4. → No contradiction → No Paradox

In each possible interpretation (meaning), there is no paradox. The paradox only exists when one falsely combines (equivocates) the two interpretations as one.

[TheVat]

Contradiction seems to arise not from statements in themselves, but because they are construed as speech acts. When you are looking at speech act that is self-referential, there is ambiguity. "Everything I say is a lie." Did you mean everything you said, up until this most recent statement, but not including it? Contradiction is a result of ambiguity. For this reason, I can admire RJG's attempts to clarify exactly what a speaker refers to.

Contradiction requires a "bad faith" moment on the part of the speaker. If the speaker uses the sentence I example and means "everything including what I'm saying now," then they are acting in bad faith and using the contradictions inherent in natural language to leave themselves a loophole. IF their most recent statement is a lie, then in fact their previous statements could possibly include some truthful ones.

[neuro]

Why does the world conform to logic?
I believe that logic tries to conform to the world: it is a human attempt at setting rules on the world so that we can deal with it.

"This statement is false" has no interest, as a statement, unless it is interpreted to state that a statement negates itself.
The same would be true for the statement "this statement is true" (again claiming its own truth).
There is no independent way of telling whether either statement is true or false, because the autoreferential statement has no premises of truth to rely on.
Which means that language permits creating statements that are neither true nor false: they simply are unverifiable claims.
Simply, not all sentences can be considered as comprised in the domain of logics. It is possible to create statements that have no value of truth because they are autoreferential bubbles.

I.e. logics cannot be applied to ALL statements. If you force logic to alìpply to ALL possible statements, then logic will become intrinsicaly liable to inconsistency.
Any mathematical system is either incomplete (it cannot encompass all domains) or inconsistent.

[Lomax]

Lots of people here are saying that "this statement is false" has no truth-value because its only referent, if any, is itself; but I still haven't had an answer to my point: what does "P = P" refer to? Or "~(P & ~P)"? We should not confuse meaning with referent. For centuries we did not know the morning star was the evening star. "John has read about the morning star" would have had a different meaning to "John has read about the evening star", despite containing all the same referents.

Suppose we had these two premises:

(i) P and Q
(ii) if P then not-Q

Neither of them refer to anything in the real world, for a start. Neither of them look like contradictions, because neither of them are. Yet we're forced to throw one of them out, aren't we? If they were both true we'd have

(iii) Q and not-Q

Well with the Liar Paradox we have

(iv) this statement is false

From this, most of us here seem to agree we can infer

(v) statement (iv) is false

and (by modus ponens, or more simply, petitio principii)

(vi) (iv) is true

In other words we get a contradiction, just like before. So why not do what we did last time around? If (iv) by itself leads to a contradiction then it is a contradiction, not an antinomy; it is just one of those rare contradictions that wears its duplicity on its sleeve. Why bring reference in to that at all? The point of formal logic is rather to avoid such things.

[neuro]

Lomax, I appreciate and agree.

My only further point is that is is not by chance that you cannot write statements like "this statement is false" using logical notation. There is no symbol for autoreference (this same statement).

So, one has to admit that there is a subdomain of language (the set of possible statements) that is not amenable to logic - logic is INCOMPLETE.

If you split the statement as you did, in order to be able to somehow deal with it using logical formalism, then you get a contradiction, as you just did.
I.e. if you try and extend the domain of logic outside its intrinsic limit, to make it COMPLETE, then it becomes INCONSISTENT.

Isn't this a trivial case of the much more general Goedel's theorem about any mathematical/logical system?

[Lomax]

Perhaps, although Godel's IV theorem (the famous one) was about the axioms of the system, and referred to the system's ability to generate true statements which it could not prove, so I would say that the paradox probably falls outside of this domain.

As for whether it can be rendered by logical notation, I think that depends on the logic. In Fregean/Aristotlean propositional logic we can simply render it as

(i) S

as RJG did earlier. We might also render it as

(i) ~i

As to whether it can be rendered in predicate logic, I think that would be tougher. We might say we can render it simply as

(i) ExFx

Where the predicate F means "asserts ~ExFx", but perhaps some would argue we've split the statement into two, and changed the nature of the game.

[TheVat]

"Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it."

— Wittgenstein, Section 4.121, Tractatus


Carnap answers:

"Philosophy is to be replaced by the logic of science – that is to say, by the logical analysis of the concepts and sentences of the sciences, for the logic of science is nothing other than the logical syntax of the language of science."

— Carnap, Foreword, Logical Syntax of Language

"According to this view, the sentences of metaphysics are pseudo-sentences which on logical analysis are proved to be either empty phrases or phrases which violate the rules of syntax. Of the so-called philosophical problems, the only questions which have any meaning are those of the logic of science. To share this view is to substitute logical syntax for philosophy."

— Carnap, Page 8, Logical Syntax of Language

Do the paradoxes being discussed also "violate the rules of syntax"?

[Asparagus]

@BIV
Catnap was expressing anti-realism, right? Metaphysicians are speaking as if their statements are truth-apt. Catnap says this is a mistake. The paradoxes we've looked at were asking questions whose answers appear to be: if yes, then no. It's a different breed of problem than the one ontological anti-realism is addressing. Agree?